Step by Step
1
The two core forms L'Hôpital directly handles
L'Hôpital's Rule directly applies to 0/0 and ∞/∞ forms — differentiate numerator and denominator separately and re-evaluate, repeating if still indeterminate.
2
Rewriting 0 × ∞
If a limit gives the indeterminate form 0 × ∞ (one factor going to 0, the other to infinity), rewrite it as a fraction first — put one factor in the denominator as its reciprocal — turning it into a 0/0 or ∞/∞ form that L'Hôpital's Rule can then handle directly.
3
Handling exponential indeterminate forms
For the forms 1^∞, 0⁰, or ∞⁰, take the natural log first. This converts a troublesome exponent into a multiplication problem, which can then often be rewritten as a fraction and handled with L'Hôpital's Rule.
4
Never skip the verification step
Regardless of which indeterminate form you're facing, always confirm you actually have one of these recognized indeterminate patterns before applying any of these techniques — applying L'Hôpital's Rule (or log tricks) to a limit that isn't actually indeterminate produces an incorrect result.
Applied Walkthrough
1
Evaluate lim(x→0+) x·ln(x). Direct substitution gives 0 × (−∞) — the indeterminate form 0 × ∞.
2
Rewrite as a fraction: x·ln(x) = ln(x) / (1/x). Now direct substitution gives −∞/∞, a form L'Hôpital's Rule can handle directly.
3
Differentiate numerator and denominator separately: d/dx[ln(x)] = 1/x, and d/dx[1/x] = −1/x². The new ratio is (1/x) / (−1/x²) = −x.
4
Take the limit: lim(x→0+) −x = 0. So the original limit is 0.
Exam Application
Exams test whether you can recognize and correctly rewrite the less-common indeterminate forms (0×∞, 1^∞, 0⁰, ∞⁰) into a 0/0 or ∞/∞ form before applying L'Hôpital's Rule.
⚠ Common Trap
The most common trap is trying to apply L'Hôpital's Rule directly to a 0×∞ or exponential indeterminate form without first rewriting it as a fraction (or taking a log) — the rule only works directly on 0/0 or ∞/∞.
✓ Quick Self-Check
1. What are the two forms L'Hôpital's Rule applies to directly?
0/0 and ∞/∞.
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2. How do you handle a 0 × ∞ indeterminate form?
Rewrite it as a fraction (putting one factor as a reciprocal in the denominator) to convert it into a 0/0 or ∞/∞ form.
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3. How do you handle 1^∞, 0⁰, or ∞⁰ indeterminate forms?
Take the natural log first, which converts the exponent into a multiplication problem that can then often be rewritten as a fraction.
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4. Evaluate lim(x→0+) x·ln(x).
0.
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5. What must you always verify before applying any of these techniques?
That the limit is genuinely one of the recognized indeterminate forms — applying these techniques to a non-indeterminate limit gives an incorrect result.
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